Effective Width Concept

11

1122

y

xme

bb

Etb Y

Effective width, long plate

Plate slenderness

p

Effective width, short plate ba ,

aa

22Y

yme

9.019.19.0

Post buckling capacity of platesInfluence of boundary conditions

B

AC

DF

E

girder stiffner

bbbbbbb

Boundary conditions depend on location of plate element

Classification example: Ship bottomA: unrestrained

B: constrained

F: restrained

Post-buckling capacity of platesInfluence of in-plane restraint of long edges

b - 2t2t 2t

Y

r

tb

Idealized

Real

Tension

Compression

2tb

2Y

r

Compressive stress 5.2112

2

211

,

tbE

ER t

Y

rr

11

18.08.1

bb

bb

e

2Y

xue

Reduction factor

DnV Class. Note 30.1

Buckling of cylindrical shells

L

B

1

1

CL

CL

Imperfect shell

Perfect shell

• Bifurcation point buckling:change from stable pre-buckling path to unstable post-buckling path

• Limit point buckling:due to imperfection sensitivity

Buckling modes Shell buckling; Buckling of shell plating between

stiffeners and frames. Interframe shell buckling; Involves buckling of the longitudinal

stiffener with associated shell plating. Panel ring buckling; Buckling of rings with associated plate

flange between longitudinal stiffeners. General buckling; Involves bending of shell plating,

longitudinal stiffeners as well as ring frames.

Torsional or local buckling of stiffeners and frames.

Column buckling of the cylinder.

zy

x

Ring frame

Longitudinalstiffner

s

l

l

l

L

r

Equilibrium equations

rNx

Nx x

0

rNx

Nx

0

4x

2

2 x

2

2

2

2w1D

p Nwx

2r

Nw

x1r

Nw 1

rN

Donnel’s equation

8wD

Nwx

2r

Nw

x1r

Nw Et

Drwx

4

x

2

2 x

2

2

2

2 2

4

4

Stress analysis – beam theory

pQ

M

N TQ

M

NT

Stress analysis – ring stiffened cylinders

w

x

R

p (<0)

l/2l/2

b

t

l

yielding

Stress Distribution Over a Half-Ring Frame.

0

0,2

0,4

0,6

0,8

1

1,2

0 0,1 0,2 0,3 0,4 0,5

Loaction between ring x /(l /2)

Hoo

p st

ress Stress in unstiffened cylinder

Stress at ring stiffener

Midway between rings At ring stiffener

3

Donnel’s equation

8wD

Nwx

2r

Nw

x1r

Nw Et

Drwx

4

x

2

2 x

2

2

2

2 2

4

4

8 4 04 2

4 22

Et w P wD w2 rx xr

Axial compression

Euler buckling stress – axial compression

xEE t

lm + n

mZ m

m + n

2

2

2 2 2 2

2

2

4

2

2 2 212 112

Zlrt

2

21 Batdorff parameter

xE 2 clE t

lZ

Etr

2 2

212 14 3

0 605. Minimum

The buckling coefficient for various modes

1

10

100

1000

10000

1 10 100 1000

Batdorf parameter Z

Buc

klin

g co

effic

ient

m,n

1,4

1,3 & 2,43,4 4,4

2,3 4,3 3,31,2 & 4,23,2 4,12,23,12,11,11,0

Approximate buckling coefficient

Buckling coefficients for axial compression

22

22

1

1

rtsZ

rtlZparameterBatdorf

s

Buckling coefficient

Approximate buckling coefficient –axial compression

ZCZ

148 2

4

Buckling coefficients for external pressure

22

1rtlZparameterBatdorf

Buckling coefficient

Buckling coefficient for torsion

22

1rtlZparameterBatdorf

Buckling coefficient

Buckling of imperfect cylindrical shells

•Buckling load influenced notably by:

– Nonlinear material

– Shape imperfections

•Scatter in test results

•Design curve lower bound

Influence of shape imperfection on buckling coefficient

• Plates moderately sensitive to imperfections

• Shells sensitive to impercetions

2

1

C

Classical theoryImperfect shell

rtZ

221

1

10

C

100

10 100 Z 1000

Plate factor

Shell factorImperfection factor

Curved panels – shell buckling

Axial Stress

4

0 702. ZS 05 1

150

0 5

..

r

t

Shear Stress 5 34 4

2

.

sl

0 856 3 4.sl

ZS

0.6

Circumfrencial Compression 1

2 2

sl

104.sl

ZS

0.6

Unstiffened cylindrical shell buckling

Buckling Coefficients For Unstiffened Cylindrical Shells, Mode a) Shell Buckling Axial Stress

1

0.702 Z 05 1

150

0 5

..

r

t

Bending

1

0.702 Z 0 5 1

150

0 5

..

r

t

Torsion and Shear force 5.34 0856 3 4. Z 0.6 Lateral Pressure 4 104. Z 0.6 Hydrostatic Pressure 2 104. Z 0.6